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Answer :
Final answer:
The question is asking about standard distribution in statistics, specifically related to the height of adults. We use z-scores to find the proportions of adults falling within specific height ranges or under a particular height h. This involves standardizing values and using the z-table, a part of the standard distribution theory.
Explanation:
The question is about the standard distribution, which is a concept in statistics. The standard distribution is used to estimate the probability of an event within a seen data set. Given the height of adults follows a normal distribution with a mean of 175 cm and variance of 16 cm2.
a) To find the proportion of adults taller than 181 cm, we need to standardize the value of 181 cm, which is the process of converting it into z-score. In a standard distribution, the z-score of a particular data point is given by z = (x-μ)/σ where x is the data point, μ is the mean and σ is the standard deviation. In this case, our standard deviation would be the square root of the given variance, which is 4 cm. The z-score for 181 is (181-175) / 4 = 1.5. Looking this score up on the z-table, we find the corresponding proportion under the curve, approximately 0.0668 or 6.68% of the adults are taller than 181 cm.
b) To find the proportion of adults whose height lies from 163 cm to 179 cm, we standardize both these values and find respective z-scores are -3 and 1. Looking these scores up on the z-table, we find the areas under the curve to their left are 0.0013 and 0.8413 respectively. Keep in mind that the area under the standard curve represents proportion. By subtracting the two proportions, we find approximately 0.84 or 84% of the adults have heights between 163 cm and 179 cm.
c) If 33% of adults have a height less than h, we look up 0.33 in the z-table to find the corresponding z-score which is approximately -0.44. Using the formula h = z * σ + μ, we find h = -0.44 * 4 + 175 = 173.24 cm. Thus, the height h which 33% of adults have height less than it is about 173.24 cm.
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Final answer:
The question is about the Normal distribution in statistics. It asks for various calculations based on the given average and variance of adult height. Answering requires knowledge of z-scores and z-tables.
Explanation:
The questions refer to the concept of Normal distribution in statistics. In this case, the average adult height is 175 cm, with a variance of 16 cm2. You're required to solve three tasks: compute the ratio of adults taller than 181 cm; calculate the ratio of adults whose height is between 163 cm and 179 cm; and find the height below which 33% adults fall.
This type of problem is usually solved by converting the height to a standardized score or a z-score, followed by using a z-table to find the related probabilities.
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